INTEGRASI NUMERIK
Pegtr Pegtegrl umerk merupk lt tu r yg dguk ole lmuw utuk memperole jw mpr proksms dr pegtegrl yg tdk dpt dselesk ser ltk. Msly dlm termodmk, model Deye utuk megtug kpsts ps dr ed pdt.
INTEGRASI NUMERIK Fugs yg dpt dtug tegrly : Fugs yg rumt msl : os e.5s.5 d d C e e d C s d os C os d s C d l l d C l C
INTEGRASI NUMERIK Pertug tegrl dl pertug dsr yg dguk dlm klkulus, dlm yk keperlu. dguk utuk megtug lus der yg dts ole ugs y = d sumu. Peerp tegrl : megtug lus d volume-volume ed putr
Dsr Pegtegrl Numerk Pejuml eroot dr l ugs... d -
Dsr Pegtegrl Numerk Melkuk pegterl pd g-g kel, sepert st wl eljr tegrl pejuml g-g. Metode Numerk y meo utuk le ept d le medekt jw eksk. 8 6 5 7 9 5
Formul Newto-Cotes - Berdsrk pd d d I Nl mpr deg poloml Dsr Pegtegrl Numerk
s ugs ler s ugs kudrt
s jug ugs kuk tu poloml yg le tgg
Poloml dpt ddsrk pd dt
INTEGRASI NUMERIK us der yg drsr dpt dtug deg : = d
Metode Itegrl Rem.5.5 *os**ep-*+.5 *os**ep-*+.5..5..5..5.5.5
Metode Itegrl Rem us yg dts y = d sumu us dg mejd N g pd rge = [,] Kemud dtug : lus setp perseg pjg dm =.
Metode Itegrl Rem us keseluru dl juml d dtulsk : Dm Ddpt..... d...
Coto = d Htug lus yg dts y = d sumu utuk rge = [,] **.8.6.......5.6.7.8.9
Coto Deg megml =. mk dperole tel :.....,85, 85.9 Ser klkulus :.6.5.6.9.6 d,....8. Terdpt kesl e =,85-, =,5
Algortm Metode Itegrl Rem: Desk ugs Tetuk ts w d ts t tegrs Tetuk juml pemg re N Htug =-/N Htug. N
Metode Itegrs Trpezod Aproksms grs lurus ler d
Coto: Atur Trpesum Htug tegrl dr Solus eksk e d e d Atur trpesum e e e 56.9677 I e 8 d e 56. 96 87. 66 57. % 56. 96 87. 66
Atur Komposs Trpesum d d d d
Metode Itegrs Trpezod tu.....
Algortm Metode Itegrs Trpezod Desk y= Tetuk ts w d ts ts tegrs Tetuk juml pemg Htug =-/ Htug
Atur Komposs Trpesum s d uto = emple % =, = p =.^.*s*;
Atur Komposs Trpesum» =; =p; d=-/;» =:d:; y=emple;» I=trp'emple',,, I = -.797e-5» I=trp'emple',,, I = -.9e-5» I=trp'emple',,, I = -.8758» I=trp'emple',,,8 I = -.6785» I=trp'emple',,,6 I = -.87» I=trp'emple',,, I = -.989» I=trp'emple',,,6 I = -.98» I=trp'emple',,,8 I = -.98» I=trp'emple',,,56 I = -.96» I=trp'emple',,,5 I = -.97» I=trp'emple',,, I = -.98» Q=qud8'emple',, Q = -.98 MATAB uto
s d = I = -.9 e-5 Et = -. 98
s d = I = -.8758 Eksk = -. 98
s d = 8 I = -.6785 Eksk = -. 98
s d = 6 I = -.87 Eksk = -. 98
Htug tegrl dr d e I %......., %......., %.., %.., %.., 66 555 95 75 5 5 5 I 5 6 5 576 76 5 5 5 5 I 5 8 9 7 788 79 I 75 I 57 87 66 I Atur Komposs Trpesum
Atur Komposs Trpesum» =:.:; y=emple;» =::; y=emple;» =::; y=emple;» =::; y=emple;» =:.5:; y=emple;» H=plot,y,,y,'g-*',,y,'r-s',,y,'-o',,y,'m-d';» seth,'ewdt',,'mrkersze',;» lel''; ylel'y'; ttle' = ep';» I=trp'emple',,, I =.88e+» I=trp'emple',,, I =.e+» I=trp'emple',,, I = 7.888e+» I=trp'emple',,,8 I = 5.768e+» I=trp'emple',,,6 I = 5.559e+
Atur Komposs Trpesum I e d
Atur Smpso / Aproksms deg ugs prol d
d d let,,,, Atur Smpso /
ξ ξ ξ ξ ξ ξ dξ ξ ξ dξ ξ dξ ξ ξ dξ d d Atur Smpso /
Atur Komposs Smpso... - -
Metode Itegrs Smpso Deg megguk tur smpso, lus dr der yg dts ugs y= d sumu X dpt dtug seg erkut: tu dpt dtulsk deg:... gep gjl N = = + + 5 +... +
Cr II Buku Rld Mur Polom terpols Newto-Gregory derjt yg mellu ketg ttk ts!! p
Cr II Buku Rld Mur Itegrsk p pd selg [,] 6 8 6! d d p d
Cr II Buku Rld Mur Meggt Mk seljuty
Atur Smpso /8 Aproksms deg ugs kuk 8 d
8 ; d d Error Pemeggl ; 68 8 E 5 5 t Atur Smpso /8
Metode Itegrs Guss Metode Newto Code Trpezod, Smpso erdsrk ttk dt dskrt. Deg ts : H sm us dtug dr smp Megktk error yg dslk ukup esr.
Metode Itegrs Guss Msl megtug us deg metode trpezod deg selg [-,] I d Persm dpt dtuls dseut pers Kudrtur Guss I d Msl =-, = d = = mejd m. trpezod Kre,,, d semrg mk kt rus meml l terseut segg error tegrsy m
Metode Itegrs Guss Bgm mer,,, d Persm dw dggp memeu ser tept l empt polom erkut djdk ugs tegrl pd tervl tegrs [-, ] = ; = ; = ; = d I d d d d Ddpt
Metode Itegrs Guss Persm dw dmk metode Guss egedre ttk d
Trsorms d Rge [,] [-,] X u gu d du g u du
Trsorms du d u u u u u u - u
Trsorms du u du u g du u g u u g
Als Ddgk deg metode Newto-Cotes Trpezod, Smpso /, /8 metode Guss- egedre ttk le seder d ese dlm opers rtmtk, kre y memutuk du u evlus ugs. e telt ddgk deg metode Newto- Cotes. Nmu kd rus metrsorms terle dulu mejd g u du
Algortm Itegrs Kudrtur Guss deg Pedekt ttk Desk ugs Tetuk ts w d ts ts tegrs Htug l kovers vrel : Tetuk ugs gu deg: Htug u u u g g g
Coto Sol
Metode Guss egedre Ttk Prmeter,,,, d dpt dr deg memut pelr w kudrtur Guss erl tept utuk 6 u ugs erkut : Deg r yg sm ddpt d I 5 ; ; ; ; 5 ; 5; 9 5 ; 9 8 ; 9 5
Metode Guss egedre Ttk 5 9 5 9 8 5 9 5 g g g du u g
Algortm Metode Itegrs Guss Deg Pedekt Ttk
Metode Guss -Ttk
Beerp Peerp Itegrs Numerk Megtug us Der Berdsrk Gmr Megtug us d Volume Bed Putr
Megtug us Der Berdsrk Gmr 9 6 5 Skl : Utuk megtug lus tegrl d pet d ts, yg perlu dlkuk dl med tu memut grs grd pd setp step stu yg dytk dlm stu kotk. Bl stu kotk mewkl mm, deg skl yg terter mk errt pjgy dl. mm tu m. Pd gmr d ts, mul ss kr deg grd ke d ss k grd ke dlm l =. Tgg pd setp grd dl seg erkut: 5
Megtug us Der Berdsrk Gmr Dr tel d ts, lus re dpt dtug deg megguk mm metode: Deg megguk metode tegrs Rem Deg megguk metode tegrs trpezod 6 y 7.5 5 y y6 y 7.5 Deg megguk metode tegrs Smpso 6 y y y y gjl gep 7
Megtug us d Volume Bed Putr us ed putr: d Volume ed putr: p V p d
Coto : 5 m 7 m I II III IV m 6 m m 7 m stu dlm m Rug ed putr dpt dedk mejd g g I d III merupk etuk slder yg tdk perlu dtug deg memg-g keml rugy, g II d IV perlu dpertugk keml. Bg I: 7 56 I Bg III: V I III V III 7 96 88 78
Coto : Sedgk utuk megtug g II d IV dperluk pemg re, mslk deg megml = dperole: Pd g II d IV: II IV d VII VIV Deg megguk tegrs trpezod dpt dperole: V II II IV 5 8 y y y VIV y y5 y 87. 5
Coto : us permuk dr otol dl: us = 758. m Volume otol dl: Volume = 98.86 m V I II III 56 8 56 758. V I V II V III IV 88 8 V 96 87.5 78 87.5 99 IV